Reproducibility variance

Diesel cetane number reproducibility concerns usually develop whenever motor cetane numbers differ by more than one number. The reproducibility variation arises due to operator and engine differences. The typical reproducibility variance accepted by ASTM will change with increasing cetane number. The following reproducibility and repeatability limits have been established by ASTM for method D-613. 

This is an example of three-way analysis of variance with no design-point replication. As we have only one value for each set of factors, the variance or the mean square within the cell as an estimate of system variance cannot be calculated. In the lack of error variance, or rather Reproducibility variance. Interaction of a higher order can be used as error estimate for the F-test. Although all statisticians do not agree with this approach, the three-way interaction variance C x R x L was taken as the error estimate for F-test. The tabular results show that only the effects of columns and layers, or temperature and catalyst, are significant. Pressure and interaction are not important at the 95% confidence level. The other approach in estimating repro-... 

An increase in the number of replicated trials causes a decrease in reproducibility variance or experimental error as well as in the associated variances of regression coefficients. Design points-trials can be replicated in all points of the experiment or in some of them. An upgrade of the design of experiment may be realized by a shift from fractional to full factorial experiment, a switch to bigger replica (from 1/6 to 1 /2 replica), a switch to second-order design (when the optimum region is dose by), etc. 

Based on the outcomes in the experimental center, these reproducibility variances have been determined Sp =4.466 with this degree of freedom f=n0-l=5. Regression coefficients have these values ... 

The reproducibility variance from five replicated design points has the value Sp =60.93 or Sp=7.81. 

The basic levels and variation intervals are given in Table 2.170. It is known from previous design points that the optimum is within the studied factor space. Orthogonal design has therefore been done to obtain the regression model, Table 2.171. Reproducibility variance of the experiment is determined from four additional design points in the experimental center (y01=61.8 y02=59.3 y03=58.7 and y04=69.0). 

Six design points in the experimental center had to be done due to Table 2.164. For the sake of economy only one design point in the experimental center was set in this example, since the reproducibility variance was obtained in the basic experiment. By processing all 15 design points, the following regression coefficients for second-order model were obtained ... 

Where the reproducibility variance is Sy=375.0, the arithmetic value of Fisher s criterion is Fr=1.21. 

Variance homogeneity facilitates calculation or estimate of experiment variance-reproducibility variance (oyj, which characterizes the total experiment error. Knowing (Sy) is necessary for regression analysis. Calculation of reproducibility variance depends on conditions for replicating trials. In the case of variance homogeneity of replicated trials and the same number of replications, reproducibility variance is determined thus ... 

SSE-is sum of squares of reproducibility variance fE-is degree of freedom of reproducibility variance. 

CCRD for k=3 has been analyzed in Example 2.44. The design included twenty trials, six of which in null point (six replications under identical conditions in the experimental center), eight FUFE and six trials in starlikC points (Table 2.139). The reproducibility error or reproducibility variance was determined from replications in the design center. 

In the case of replications of only FUFE 2s in the first phase when the basic experiment was defined, this value of reproducibility variance was obtained Sy = 0.32. With 95% confidence it is evident that the obtained value does not statis-... 

The arithmetic value of the F-criterion calculated by formula (2.160), is compared to its tabular value (Table E) for the chosen significance level and the associated degrees of freedom in order to check the statistical significance of the difference between the lack of jit variance and reproducibility variance. When this difference is sta-... 

Besides the number of trials as defined by the choice of design of experiments, it is important to determine the number of replicated trials. Replications are necessary to eliminate robust errors and to determine the reproducibility variance or error of experiment. Since the reproducibility variance has in this case been quite reliably determined in previous study, we may therefore accept a minimal number of replications or a single replication (Sy2=1.0). Prior to doing an experiment one should define the sequence of performing trials, which should be random to annul systematic errors or outside effects. By means of a table of random numbers this sequence has been chosen 15 13 10 5 14 4 6 1 7 8 3 2 9 12 11 and 16. The outcomes of these experiments are given in Table 2.231. 

Homogeneity of replicate variances makes calculation of reproducibility variance possible ... 

Variances of replicated design points are equal, so that the reproducibility variance is ... 

The repeatability, within-laboratory reproducibility, and reproducibility variance are derived from ... 

The next step in developing a statistical model is the verification of the significance of the coefficients by means of the Student distribution and the reproducibility variance. 

When the homogeneity of the variances has been tested, we continue to compute the values of the reproducibility variance with relation (5.64) ... 

In statistics, the reproducibility variance is a random variable having a number of degrees of freedom equal to u = N(m — 1). Without the reproducibility variances or any other equivalent variance, we cannot estimate the significance of the regression coefficients. It is important to remember that, for the calculation of this variance, we need to have new statistical data or, more precisely, statistical data not used in the procedures of the identification of the coefficients. This requirement explains the division of the statistical data of Fig. 5.3 into two parts one sigmficant part for the identification of the coefficients and one small part for the reproducibility variance calculation. 

After calculating the value of the random variable F, we establish the reproducibility variances and carry out the test according to the procedure given in Table 5.6. Exceptionally, in cases when we do not have any experiment carried out in parallel, and when the statistical data have not been divided into two parts, we use the relative variance for the mean value (s ) instead of the reproducibility variance. This relative variance can be computed with the statistical data used for the identification of the coefficients using the relation (5.64) ... 

The use of the reproducibility variance allows the significance test of the coefficients of the final regression relationship ... 

N gives the total number of experiments in the plan. When we use a complete second order plan, it is not necessary to have parallel trials to calculate the reproducibility variance, because it is estimated through the experiments carried out at the centre of the experimental plan. The model adequacy also has to be examined with the next procedure ... 

In the table, the differences between the columns result from the change in the values of the factors and the differences between the lines give the reproducibility problems of the experiments. The total variance (s ) associated to the table data, here given by relation (5.151), must be divided according to its components the variances of inter-lines (or reproducibility variances) and variances of inter-columns (or variances caused by the factor). 

The real residual variance frequently named reproducibility variance can be determined by repeating all the experiments but this can turn out to be quite expensive. The Latin squares method offers the advantage of accepting the repetition of a small number of experiments with the condition to use a totally random procedure for the selection of the experiments. With the data from Table 5.61 and... 

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